A Splitting in Time Scheme and Augmented Lagrangian Method for a Nematic Liquid Crystal Problem

Abstract

We study the numerical approximation of nematic liquid crystal flows governed by a Ericksen–Leslie problem. This problem couples the incompressible Navier–Stokes dynamic with a gradient flow system related to the orientation unitary vector of molecules. First, a two sub-step viscosity-splitting time scheme is proposed. The first sub-step couples diffusion and convection terms whereas the second one is concerned with diffusion terms and constraints (divergence free and unit director field). Then, in the first sub-step we use a Gauss–Seidel decoupling algorithm, and in the second sub-step, we use Uzawa type algorithms on augmented Lagrangian functionals to overcome the divergence free and the unit director field constraints. From the computational point of view, it is a fully decoupled linear scheme (where all systems to solve are for scalar variables). Some numerical experiments in 2D domains are carried out by using only linear finite elements in space, confirming at least numerically the viability and the convergence of our scheme.